Optimum stacking pattern for multi-stream plate-fin heat exchanger through a genetic algorithm

Abstract

A methodology has been developed to determine optimal stacking pattern of multi-stream plate-fin heat exchangers using a genetic algorithm. This investigation tries to find out the stacking pattern, which gives the maximum heat load for a given number of fluid streams with some specified properties, mass flow rate and entry temperature. The method of area splitting and successive partitioning has been used for the estimation of heat load. Several direct and indirect checks have been made to test the optimality of the stacking pattern. These include exhaustive search, checking the pattern of cumulative heat load and plate temperature profile along a plane normal to the exchanger. The solutions obtained through the developed algorithm exhibit excellent conformity to all of these criteria.

Introduction

Heat exchangers are unique thermal equipments which facilitate the exchange of thermal energy among various fluid streams. As heat exchangers cover an exceedingly wide range of applications, numerous designs have been evolved over the years. Those designs not only vary in configuration and size but also have different working principles. Historically, two stream exchangers were conceived first and till today, the majority of exchangers belong to this category. Nevertheless, to cope with the continuous demand for high compactness, low capital and running costs, the engineers thought of encasing more than two streams in a single envelope. Such a device is termed as a multi-stream heat exchanger, where the streams (or at least one of the streams) will have more than one thermal communication. Various designs of multi-stream heat exchangers are in use. The simplest configuration could be a concentric tube three-stream heat exchanger, in which only the intermediate fluid stream will have more than one thermal communication. Besides this, shell and coiled tube type, plate and frame type and plate-fin type are the main varieties available. In many of these varieties, there is a limitation as far as the number of streams is concerned. For example, shell and helical exchangers are able to handle one cold and two or more hot streams or vice-versa.

In this respect, plate-fin heat exchangers enjoy a great flexibility, as there is no restriction on the number of hot and cold streams. A plate-fin heat exchanger with as many as ten or twelve fluid streams is not uncommon in industry. Plate-fin heat exchangers offer a number of advantages. For example, with a small volume and weight, it offers a high thermal performance. The pressure drop through such units is moderate, while the approach temperature could be very close. The possibility of parallel or cross arrangement of fluid streams and their intermediate entry and exit offer additional flexibility in the design of these exchangers. The main concerns behind the use of plate-fin heat exchangers for multi-fluid applications are the limited ranges of temperature and pressure at which they can operate and the restrictions regarding their applicability to relatively clean fluids. Nevertheless, these heat exchangers are widely used in cryogenics and other process plants.

Over the decades, the analysis and design of two-stream heat exchangers got standardized. Both theoretical models as well as computational algorithms are available even for handling the cases of variable properties of the fluids, non-uniform heat transfer coefficient, phase change, axial conduction, axial dispersion etc. Unfortunately, the design procedure of multi-stream heat exchangers is still under development due to its inherent more complexity compared to the design of two-stream exchangers.

Features like bypass heat transfer [1] or crossover in temperature [2], which are common in multi-stream heat exchangers, have no equivalence in two-stream units. The unique parameters like effectiveness, Number of Transfer Units (NTU) or Log Mean Temperature Difference (LMTD) have simplified the analysis and design of two-stream heat exchangers. The designers are yet to find out equivalence of such parameters in case of multi-fluid heat exchangers. A brief appraisal of various design methodologies suggested for the plate-fin heat exchangers is given here. In the simplest form, a multi-stream heat exchanger consists of three different streams. Sorlie [3] developed a design theory for three-fluid heat exchangers of the concentric-tube and plate-fin types, in which the intermediate and cold streams were thermally insulated. Aulds and Barron [4] extended the work of Sorlie [3] by analyzing the case, in which all three streams were in thermal communication, as it is relevant for many three-fluid heat exchangers used in cryogenic systems. Pioneering work on multi-channel, parallel flow heat exchangers was carried out by Kao [5] and Wolf [6]. It had been shown by them that in the absence of the effect of axial conduction through the separating wall, the basic equations describing the process of heat transfer in a multi-channel heat exchanger are a set of linear differential equations involving the temperatures of the fluids and the separating walls. A similar approach had been adopted by Zaleski [7] to analyse multi-channel heat exchangers with unconnected channels, particularly lamella-type, plate-type, and helix-type units. Chato et al. [8] suggested a method of dividing the heat exchanger into a large number of smaller sections over which physical properties remain approximately constant. Haseler [1] proposed a novel solution methodology termed as constant wall temperature assumption. Based on this, the temperatures of all the separating walls were assumed to be equal at any cross-section normal to the flow direction. Prasad and Gurukul [9], [10], in their formulation of the differential method for design of plate-fin heat exchangers, applied the same simplified idealization. Prasad [11], [12], [13] employed the modified shooting method to solve the governing equations. Luo et al. [14] developed an analytical model of a multi-stream exchanger with constant physical properties. In a separate paper, Luo et al. [15] proposed a more generalized analytical solution for predicting the thermal performance of multi-stream heat exchangers and their networks.

Pinch Technology is a method usually adopted for the analysis of heat exchanger networks. Polley and Picon-Nunez [16], Picon-Nunez and Polley [17] and Picon-Nunez et al. [18] extended this technique for multi-stream plate-fin heat exchangers using the temperature vs. enthalpy diagrams or composite curves. Wang and Sunden [19] presented a new methodology for design of multi-stream plate-fin heat exchangers through optimization of heat exchanger networks. Ghosh et al. [20] suggested an alternate algorithm for the analysis of multi-stream plate-fin heat exchangers. Two key concepts had been used by them. The multi-stream heat exchanger had been conceived as a combination of a number of overlapping two-stream heat exchangers. This needs apportioning the heat exchanger area between different streams, which had been achieved by Area Splitting Method. Next, the heat exchanger had been progressively subdivided in the axial direction by Successive Partitioning Method to improve the accuracy of prediction.

As the thermal engineers always face the challenge of improving the performance of heat exchangers under constraints of varied nature, various efforts have been made to optimize the design of plate-fin heat exchangers. One of the earliest efforts of optimizing the stream arrangements of a multi-stream plate fin heat exchangers is due to Fan [2]. He recommended a methodology for finding out the optimum arrangement of streams through the fragmentation of individual streams. In this context, clearly specified recommendation has been prescribed by Suessman and Mansour [21]. They proposed a technique for arranging the fluid streams, such that the resulting stacking pattern becomes close to the optimum if not the optimum. According to them, for an optimum stacking pattern, the value of the cumulative heat load should change its sign as one move from one stream to the next. It may be noted that Prasad [22] has also used the technique of Suessman and Mansour [21] for studying the layer stacking pattern in multi-stream heat exchangers. However, Prasad [22] has used the concept of “half fin idealization” while analyzing multi-stream exchangers and studying the layer stacking pattern effect. The idealization used in the paper was quite weak. Later on Prasad [12], [13] has been successful in formulating the fin equations of multi-stream heat exchangers in a generalized way and solving them iteratively.

On the other hand, Yuan et al. [23] suggested a different technique for designing a multi-stream heat exchanger involving two steps. In the first step (predict), the initial design of the exchanger is made through a local balance principle. In the second step (correct) the passage arrangement is readjusted using the results of differential computation of the temperature distribution. Illustration through three case studies were used to establish the feasibility of the proposed method.

Parallel efforts are also made to optimize the design of multi-stream exchangers through different programming techniques and soft computing methodologies. A multivariate optimization of plate-fin heat exchangers had been carried out by Sunder and Fox [24]. They employed non-linear programming (NLP) to design a brazed aluminum plate-fin type compact heat exchanger. A multivariate objective function was minimized subject to several constraints. An approach was developed by Peng and Ling [25] for the optimization of plate-fin heat exchangers (PFHE) using a Neural Network (NN) coupled with a Genetic Algorithm (GA). The major objectives of their PFHE design were the minimum total weight and total annual cost for a given set of constrained conditions. Total length and width of PFHE core, number of hot side layers, fin height and pitch on each side of PFHE were considered as the variables to be optimized by means of a GA combined with the Back-Propagation Neural Network (BPNN). A GA-based optimization technique had been developed for cross-flow plate-fin heat exchangers by Mishra et al. [26]. The aim of optimization was to minimize the total annual cost for a specified heat duty under given space and flow restrictions. In another paper, Mishra and Das [27] optimized the thermo-economic cost for a cross-flow offset strip-fin heat exchanger with specified heat loads under given space restrictions.

Picón-Núñez and López Robles [28] suggested a unique technique for flow passage arrangement for multi-stream heat exchanger. The main feature of their approach is uniform heat load per passage. This has been achieved by selecting suitable secondary surface and fin geometry. They have considered equal number of hot and cold passages and the number of passages allocated to a given stream is directly proportional to its heat capacity flow rate. They have also proposed a simple model for the steady state model of the multi-stream heat exchanger.

In the past decade efforts have also been made to synthesize multi-stream heat exchangers as a heat exchanger network problem (HEN). In many cases such networks have been optimized. It has been identified [29] that such synthesis problems with a large number of streams may have more than one optimum solution. The difficulty of solving such problems compelled the researcher to adopt different evolutionary techniques like a combination Genetic Algorithm and simulated annealing [29], [30], improved genetic algorithm [31] etc. Improvement in GA was required to handle the problem of mixed integer nonlinear programming [32] and to avoid the premature convergence. Luo et al. [33] proposed a hybrid scheme combining GA with simulated annealing algorithm, local optimizing strategy, structure control strategy and other strategies to improve the structural search ability of the algorithm substantially. Fieg et al. [34] used a two stepped procedure for the synthesis of a large scale heat exchanger network. In the first step, a hybrid GA is used for the entire network. In the second step, the optimization of sub-networks was achieved through a monogenetic algorithm.

The accommodation of a large number of streams in a single envelope presents a unique complexity to the designer. With the increase in the number of streams, the option of arranging the streams also increases. The specific arrangement of various streams is known as stacking pattern in case of a plate-fin heat exchanger. It is obvious that all stacking patterns will not provide with the same thermal performance. The best thermal performance can be provided by a particular stacking pattern or at the best by a limited number of patterns. Again, all theoretically possible stacking patterns may not be practically viable due to the process constraints. This constitutes a unique optimization problem for multi-stream heat exchangers, which does not have a counter part in case of a two-stream heat exchanger. The need for optimizing the stacking pattern in a plate-fin heat exchanger has been appreciated by a number of researches. Time to time, some rules of thumb or broad design guide lines have been suggested. However, to the best of the authors’ knowledge, till date no algorithm for determining the optimum stacking pattern has been reported in the literature. This has motivated the present investigation.

The methodology for optimization needs to be selected based on the nature of the problem. The present case involves optimization of a combinatorial problem and the classical derivative-based algorithms may not be suitable for its solution. One can think of applying a direct search technique, which becomes too much tedious for a large parameter space. On the other hand, gradient-based search requires the estimation of derivative and has a risk of getting stuck at a local extremum. GA [35], [36], [37] is a generic and robust search technique, which generally does not suffer from the above limitations. It provides with several near-optimal solutions, which offer the designer a sufficient flexibility.

Through this study, a methodology has been developed using a GA for determining the optimum stacking pattern of plate-fin heat exchangers. Parallel flow multi-stream plate-fin heat exchangers with co-current and counter current arrangement of fluid streams have been considered. From a given population of possible solutions, the GA can select better solutions mimicking the principle of natural selection. In the present investigation, heat duty of the exchanger has been taken as the basis of selection. A suitable design algorithm is needed for the estimation of heat duty. The basic design of the heat exchanger has been obtained through the successive partitioning algorithm of Ghosh et al. [20]. Efforts have also been made to check the validity of the GA solutions. The prediction from the GA has been compared with several direct and indirect checking for heat exchangers with a limited number of streams.

The remaining part of the paper has been organized as follows: Section 2 deals with mathematical formulation of the problem. Some case-studies have been solved in section 3. Results are stated and discussed in section 4. Some concluding remarks are made in section 5.